Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} y + 3 x^{2} w - x y z - 2 x y w + x z^{2} - x w^{2} + y^{3} + y^{2} z + y^{2} w - y z^{2} - y z w $ |
| $=$ | $3 x^{2} y + 3 x^{2} z - x y z + x y w + 2 x z w + x w^{2} + y^{3} - y^{2} w - y z w - y w^{2}$ |
| $=$ | $3 x^{2} y - 3 x^{2} z - 3 x^{2} w + 2 x y z + x y w - x z^{2} - 2 x z w + y^{3} - y^{2} z + y z w$ |
| $=$ | $3 x^{2} y + 3 x^{2} w + 2 x y z + 4 x y w - 2 x z^{2} + 2 x w^{2} + y^{3} - 2 y^{2} z + y^{2} w + \cdots - w^{3}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x^{4} z - 6 x^{3} y^{2} - 2 x^{3} z^{2} + 6 x^{2} y^{3} - 9 x^{2} y^{2} z - 4 x^{2} z^{3} + \cdots - z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{2} + x\right) y $ | $=$ | $ 3x^{6} - 9x^{5} + 11x^{4} - 8x^{3} + 11x^{2} - 9x + 3 $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 108 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3^2\,\frac{1146617856xy^{21}+52744421376xy^{18}w^{3}+219290664960xy^{17}w^{4}-356024844288xy^{16}w^{5}-2750807900160xy^{15}w^{6}+16327479951360xy^{14}w^{7}+37714930753536xy^{13}w^{8}-238318286635008xy^{12}w^{9}-379815503099904xy^{11}w^{10}+3481982838650880xy^{10}w^{11}+3747864068384256xy^{9}w^{12}-45336223040965632xy^{8}w^{13}-38048172146780544xy^{7}w^{14}+555478688917090944xy^{6}w^{15}+416161083266087904xy^{5}w^{16}-6624099559144000512xy^{4}w^{17}-4963079360483292600xy^{3}w^{18}+78086869752967286040xy^{2}w^{19}+63270280980410115798xyw^{20}+1769553920xz^{21}+19092350976xz^{20}w-544175769600xz^{19}w^{2}-4375414525952xz^{18}w^{3}+26996804047872xz^{17}w^{4}+231987067130880xz^{16}w^{5}-512051787992064xz^{15}w^{6}-5294803994438400xz^{14}w^{7}+6512440090044096xz^{13}w^{8}+72546603041740160xz^{12}w^{9}-76500025212909504xz^{11}w^{10}-684471392134280640xz^{10}w^{11}+854582085101586160xz^{9}w^{12}+4664585519069731920xz^{8}w^{13}-7914216292339475160xz^{7}w^{14}-22328091637833240168xz^{6}w^{15}+54291954229772949144xz^{5}w^{16}+63807928722386123238xz^{4}w^{17}-239472532256215212967xz^{3}w^{18}-65981609779000740735xz^{2}w^{19}+376411821708716487468xzw^{20}+81584067985335167329xw^{21}+48731258880y^{19}w^{3}-79116632064y^{18}w^{4}-328935997440y^{17}w^{5}+2080681426944y^{16}w^{6}+6216352874496y^{15}w^{7}-35122624856064y^{14}w^{8}-70455967617024y^{13}w^{9}+537108115083264y^{12}w^{10}+693470669939712y^{11}w^{11}-7368680811276288y^{10}w^{12}-6899055339822336y^{9}w^{13}+92578446012162816y^{8}w^{14}+72359663584815936y^{7}w^{15}-1116325482194859264y^{6}w^{16}-826731697147174800y^{5}w^{17}+13220897842015743888y^{4}w^{18}+10222704735599376324y^{3}w^{19}-155424264652224884664y^{2}w^{20}-524201984yz^{21}-18426832896yz^{20}w+117663283200yz^{19}w^{2}+2888667127808yz^{18}w^{3}-2415161576448yz^{17}w^{4}-133183093810176yz^{16}w^{5}-73006591879680yz^{15}w^{6}+2901176868840192yz^{14}w^{7}+2355027644885184yz^{13}w^{8}-39953404443557312yz^{12}w^{9}-25747047366413376yz^{11}w^{10}+396968336413116672yz^{10}w^{11}+95510697614591792yz^{9}w^{12}-2992756690164181824yz^{8}w^{13}+881265328569956664yz^{7}w^{14}+16944020606449652136yz^{6}w^{15}-15176402702305955316yz^{5}w^{16}-65964500188894058562yz^{4}w^{17}+97923182972369908453yz^{3}w^{18}+133202882308151186340yz^{2}w^{19}-170189212409605883913yzw^{20}-88762329032128545883yw^{21}+284041216z^{22}-973963264z^{21}w-96769296384z^{20}w^{2}-272758700032z^{19}w^{3}+5011920793600z^{18}w^{4}+20049648814080z^{17}w^{5}-95974177652736z^{16}w^{6}-447376536026112z^{15}w^{7}+1134778524533568z^{14}w^{8}+5493847140383680z^{13}w^{9}-10577895963577024z^{12}w^{10}-44430862509818880z^{11}w^{11}+85240516989653648z^{10}w^{12}+249885737333539264z^{9}w^{13}-570475794570948144z^{8}w^{14}-938305301972909016z^{7}w^{15}+2802187408983594816z^{6}w^{16}+2131229646485619912z^{5}w^{17}-7675979583048256163z^{4}w^{18}-8452524646376490358z^{3}w^{19}+12849582263184480585z^{2}w^{20}+22890706751536824140zw^{21}+19778531745605513815w^{22}}{2579890176xy^{18}w^{3}+11609505792xy^{17}w^{4}-17414258688xy^{16}w^{5}-53532721152xy^{15}w^{6}+21767823360xy^{14}w^{7}+130606940160xy^{13}w^{8}+371625117696xy^{12}w^{9}-249604374528xy^{11}w^{10}-4860482858496xy^{10}w^{11}-304285953024xy^{9}w^{12}+49027283081856xy^{8}w^{13}+19344112277760xy^{7}w^{14}-487206953200800xy^{6}w^{15}-347372149618944xy^{5}w^{16}+4940530470373464xy^{4}w^{17}+5132621809538568xy^{3}w^{18}-51124291397828334xy^{2}w^{19}-70311831676080894xyw^{20}-5184xz^{21}+140832xz^{20}w+407808xz^{19}w^{2}-52335936xz^{18}w^{3}+814189320xz^{17}w^{4}-108549720xz^{16}w^{5}-62113658004xz^{15}w^{6}+211709125710xz^{14}w^{7}+1632713962257xz^{13}w^{8}-8773108929342xz^{12}w^{9}-18127950632649xz^{11}w^{10}+168821081064027xz^{10}w^{11}+17969100131349xz^{9}w^{12}-1880999104515984xz^{8}w^{13}+1898040823506783xz^{7}w^{14}+12695265805505013xz^{6}w^{15}-24159463472399535xz^{5}w^{16}-47938008617986482xz^{4}w^{17}+137136394595766024xz^{3}w^{18}+78877632928281039xz^{2}w^{19}-229133090428924950xzw^{20}-66684589337298834xw^{21}+2579890176y^{19}w^{3}-3869835264y^{18}w^{4}-17414258688y^{17}w^{5}+15479341056y^{16}w^{6}+52726505472y^{15}w^{7}+30474952704y^{14}w^{8}-112588019712y^{13}w^{9}-834554161152y^{12}w^{10}+152646861312y^{11}w^{11}+9041305743360y^{10}w^{12}+2092379112576y^{9}w^{13}-89730301087872y^{8}w^{14}-49360041675360y^{7}w^{15}+899886426358752y^{6}w^{16}+784637166001896y^{5}w^{17}-9221269535919504y^{4}w^{18}-11089147246140366y^{3}w^{19}+96202373598381132y^{2}w^{20}+1536yz^{21}-4320yz^{20}w-1412928yz^{19}w^{2}+32393568yz^{18}w^{3}-235955592yz^{17}w^{4}-2280424968yz^{16}w^{5}+31161589560yz^{15}w^{6}+5805729270yz^{14}w^{7}-1137510167103yz^{13}w^{8}+2257763551395yz^{12}w^{9}+19608581045691yz^{11}w^{10}-67578603194412yz^{10}w^{11}-173675541071481yz^{9}w^{12}+973124163777960yz^{8}w^{13}+539303159583981yz^{7}w^{14}-8252907130141686yz^{6}w^{15}+4264474115809764yz^{5}w^{16}+41379121822602465yz^{4}w^{17}-48920676974445495yz^{3}w^{18}-99653756972687937yz^{2}w^{19}+92413971832862592yzw^{20}+62110062644411658yw^{21}-832z^{22}+34432z^{21}w-355872z^{20}w^{2}-2271392z^{19}w^{3}+118576880z^{18}w^{4}-635170536z^{17}w^{5}-5250692820z^{16}w^{6}+41997292464z^{15}w^{7}+70224502203z^{14}w^{8}-1062483495601z^{13}w^{9}+264810475753z^{12}w^{10}+14512053543837z^{11}w^{11}-18521065877009z^{10}w^{12}-118193855129581z^{9}w^{13}+244348988514651z^{8}w^{14}+571627978839156z^{7}w^{15}-1626731837167944z^{6}w^{16}-1582130095749036z^{5}w^{17}+4946631939600866z^{4}w^{18}+6027337912816912z^{3}w^{19}-1692832449721143z^{2}w^{20}-8514703030538264zw^{21}-7298751061334812w^{22}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
18.108.2.e.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}-6X^{3}Y^{2}+6X^{2}Y^{3}+X^{4}Z-9X^{2}Y^{2}Z+6XY^{3}Z-2X^{3}Z^{2}+9XY^{2}Z^{2}+6Y^{3}Z^{2}-4X^{2}Z^{3}+6Y^{2}Z^{3}-4XZ^{4}-Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
18.108.2.e.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -3y^{2}z^{2}w-3y^{2}zw^{2}-3y^{2}w^{3}-yz^{4}+4yz^{3}w+\frac{15}{2}yz^{2}w^{2}-\frac{1}{2}yzw^{3}-yw^{4}+z^{5}-3z^{4}w-4z^{3}w^{2}+5z^{2}w^{3}+\frac{9}{2}zw^{4}+w^{5}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 81y^{6}z^{6}w^{3}+243y^{6}z^{5}w^{4}+486y^{6}z^{4}w^{5}+567y^{6}z^{3}w^{6}+486y^{6}z^{2}w^{7}+243y^{6}zw^{8}+81y^{6}w^{9}+81y^{5}z^{8}w^{2}-162y^{5}z^{7}w^{3}-\frac{2025}{2}y^{5}z^{6}w^{4}-\frac{3969}{2}y^{5}z^{5}w^{5}-\frac{4455}{2}y^{5}z^{4}w^{6}-\frac{2511}{2}y^{5}z^{3}w^{7}-\frac{567}{2}y^{5}z^{2}w^{8}+\frac{405}{2}y^{5}zw^{9}+81y^{5}w^{10}+27y^{4}z^{10}w-252y^{4}z^{9}w^{2}-135y^{4}z^{8}w^{3}+1890y^{4}z^{7}w^{4}+\frac{15687}{4}y^{4}z^{6}w^{5}+\frac{12987}{4}y^{4}z^{5}w^{6}+270y^{4}z^{4}w^{7}-\frac{7209}{4}y^{4}z^{3}w^{8}-\frac{5157}{4}y^{4}z^{2}w^{9}-\frac{729}{2}y^{4}zw^{10}-45y^{4}w^{11}+3y^{3}z^{12}-78y^{3}z^{11}w+\frac{609}{2}y^{3}z^{10}w^{2}+\frac{1395}{2}y^{3}z^{9}w^{3}-\frac{7839}{4}y^{3}z^{8}w^{4}-\frac{9981}{2}y^{3}z^{7}w^{5}-\frac{17667}{8}y^{3}z^{6}w^{6}+\frac{24237}{8}y^{3}z^{5}w^{7}+\frac{32931}{8}y^{3}z^{4}w^{8}+\frac{12423}{8}y^{3}z^{3}w^{9}-\frac{687}{4}y^{3}z^{2}w^{10}-\frac{501}{2}y^{3}zw^{11}-45y^{3}w^{12}-2y^{2}z^{13}+38y^{2}z^{12}w-75y^{2}z^{11}w^{2}-\frac{1007}{2}y^{2}z^{10}w^{3}+\frac{2845}{4}y^{2}z^{9}w^{4}+\frac{5283}{2}y^{2}z^{8}w^{5}-\frac{1821}{4}y^{2}z^{7}w^{6}-\frac{18249}{4}y^{2}z^{6}w^{7}-\frac{21753}{8}y^{2}z^{5}w^{8}+\frac{1211}{4}y^{2}z^{4}w^{9}+\frac{6353}{8}y^{2}z^{3}w^{10}+426y^{2}z^{2}w^{11}+\frac{445}{4}y^{2}zw^{12}+\frac{47}{4}y^{2}w^{13}-3yz^{14}+32yz^{13}w-\frac{103}{2}yz^{12}w^{2}-\frac{681}{2}yz^{11}w^{3}+442yz^{10}w^{4}+\frac{7397}{4}yz^{9}w^{5}+\frac{3861}{8}yz^{8}w^{6}-\frac{14331}{8}yz^{7}w^{7}-\frac{12021}{8}yz^{6}w^{8}-\frac{10389}{8}yz^{5}w^{9}-\frac{12373}{8}yz^{4}w^{10}-\frac{5327}{8}yz^{3}w^{11}-\frac{201}{8}yz^{2}w^{12}+\frac{313}{8}yzw^{13}+\frac{25}{4}yw^{14}+2z^{15}-\frac{43}{2}z^{14}w+32z^{13}w^{2}+\frac{839}{4}z^{12}w^{3}-\frac{817}{4}z^{11}w^{4}-\frac{9277}{8}z^{10}w^{5}-562z^{9}w^{6}+\frac{7287}{4}z^{8}w^{7}+\frac{10785}{4}z^{7}w^{8}+\frac{4735}{4}z^{6}w^{9}+\frac{307}{2}z^{5}w^{10}+\frac{703}{4}z^{4}w^{11}+\frac{635}{8}z^{3}w^{12}-\frac{145}{8}z^{2}w^{13}-13zw^{14}-\frac{13}{8}w^{15}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -6y^{2}z^{2}w-6y^{2}zw^{2}-6y^{2}w^{3}-2yz^{4}+8yz^{3}w+15yz^{2}w^{2}-yzw^{3}-2yw^{4}+z^{5}-z^{4}w-\frac{15}{2}z^{3}w^{2}-\frac{5}{2}z^{2}w^{3}+4zw^{4}+\frac{3}{2}w^{5}$ |
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
